Biological macromolecules (such as proteins) are flexible structures held together by a variety of stabilizing interactions. The aim of this proposa is to advance our understanding of how the three-dimensional structure and dynamics of large molecular assemblies relate to their biological functions. We propose a systematic (mathematical, algorithmic and biological) study of rigidity-based methods for simulating slow-motion conformational changes in biomolecules. Decomposing large molecules into rigid clusters leads to structures with a much smaller number of degrees of freedom. We treat them as kinematic linkages, i.e. as collections of articulated rigid bodies interconnected through various types of flexible joints. We will develop new methods for motion simulation, based on these kinematic abstractions. The essence of this approach is a substantial dimensionality reduction of the conformational space. To test and experiment with our ideas, we will develop new software for generating kinematically-realistic motions of biological macromolecules, built upon and integrated into the recently released software infrastructure KINARI (http://kinari.cs.umass.edu) developed in PI Streinu's group. We will evaluate and benchmark our models and our new methods, for accuracy and speed, against other coarse-grained models (such as Normal Mode Analysis) and will validate them on biological data. The mathematical and computational approach is to develop a rigorous deformation theory for molecular structures modeled as systems of articulated bodies, observant of the topology of their underlying configuration spaces and leading to effective simulation techniques through motions that are guided by essential kinematic constraints. This research is anticipated to enhance the general understanding of flexibility and allostery in proteins, to impact protocols for protein structure determination using low-resolution experimental data and, ultimately, to inform the rational design of new drugs based on improved understanding of protein functions as they relate to flexibility and motion.